Convex Grid Drawings of 3-Connected Plane Graphs Erik van de Pol

Contents Introduction Canonical Decomposition – Chord-path – Outer chain Algorithm for Convex Grid Drawing

Introduction Convex Drawing Grid Drawing – Straight line Grid Drawing Plane Graph – 2-connected – 3-connected – k-connected

Canonical Decomposition Canonical Decomposition is a generalization of a Canonical Ordering. Internally 3-connected Plane Graph G: – G is 2-connected and – For any separation pair {u, v} of G, u and v are outer vertices and each connected component of G - {u, v} contains an outer vertex.

Canonical Decomposition: Chord-path Definition: Chord-path in Graph G – Let G be a 2-connected plane graph – Let w 1, w 2, …, w t be the vertices appearing in clockwise order on the outer cycle of G. – Then path P in G is a chord-path of the outer cycle of G if P satisfies all the following criteria: P connects two outer vertices w p and w q, p < q {w p, w q } is a separation pair of G P lies on an inner face P does not pass through any outer edge and any outer vertex other then the ends w p and w q – Minimal chord-path: None of w p+1, w p+2, …, w q-1 is an end of a chord-path.

Canonical Decomposition: Outer chain Definition: Outer chain of graph G: – Let {v 1, v 2, …, v p }, p ≥ 3, be a set of three or more outer vertices consecutive on the outer cycle of G such that d(v 1 ) ≥ 3, d(v 2 ) = d(v 3 ) = … = d(v p-1 ) = 2, and d(v p ) ≥ 3. – We then call the set {v 2, v 3, …, v p-1 } an outer chain of G.

Canonical Decomposition Definition: Canonical Decomposition of Graph G – Let G = (V, E) be a 3-connected plane graph of n ≥ 4 vertices. – For an ordered partition Π = (U 1, U 2, …, U l ) of set V, we denote by G k, 1 ≤ k ≤ l, the subgraph of G induced by U 1, U 2, …, U k, while we denote by – inverse G k, 0 ≤ k ≤ l - 1, the subgraph of G induced by U k+1, U k+2, …, U l. All normal inverse rules apply. – Let (v 1, v 2 ) be an outer edge of G. – Then Π is a Canonical Decomposition of G (for an outer edge (v 1, v 2 ) ) if Π satisfies the following conditions:

Canonical Decomposition: Conditions (1) 1. U 1 is the set of all vertices on the inner face containing edge (v 1, v 2 ), and U l is the singleton set containing an outer vertex v n not є {v 1, v 2 }. 2. For each index k, 1 ≤ k ≤ l, G k is internally 3- connected. 3. For each index k, 2 ≤ k ≤ l, all vertices in U k are outer vertices of G k and the following conditions hold:

Canonical Decomposition: Conditions (2) a) If |U k | = 1, then the vertex in U k has two or more neighbors in G k-1 and has at least one neighbor in inverse G k when k < l. b) If |U k | ≥ 2, then U k is an outer chain of G k, and each vertex in U k has at least one neighbor in inverse G k.

Canonical Decomposition: Lemma Every 3-connected plane graph G of n ≥ 4 vertices has a canonical decomposition has a canonical decomposition Π, and Π can be found in linear time.

Algorithm for Convex Grid Drawing Let G be a 3-connected plane graph. Let Π = (U 1, U 2, …, U l ) be a canonical decomposition of G. The algorithm will add to a drawing the vertices in set U k, one by one, in the order U 1, U 2, …, U l, adjusting the drawing at every step. Rank: We say that a vertex v є U k has rank k

The End Any questions?